Calculate latitude and longitude having meters distance from another latitude/longitude point

Question

I need to calculate latitude and longitude of a given point.

I know latitude and longitude of a reference point, and an value indicating meters on x and y axis from the reference point. Starting from this data I have to find latitude and longitude of the point.

I searched for similar questions but it looks like most questions are about finding distance between two lat/long points. I need to do the contrary.

How can I do? I use Java

Answer 1

Here is the best starting point for such questions: Aviation Formulary. They have all the formulas for doing such kinds of things.

From these formulary, I created my own Java util class. It uses a lot of internal stuff, so I cannot post the actual class here but to give you some examples on how to convert the knowledge from the formulary into Java code.

Here're some basic methods:

/**
 * the length of one degree of latitude (and one degree of longitude at equator) in meters.
 */
private static final int DEGREE_DISTANCE_AT_EQUATOR = 111329;
/**
 * the radius of the earth in meters.
 */
private static final double EARTH_RADIUS = 6378137; //meters
/**
 * the length of one minute of latitude in meters, i.e. one nautical mile in meters.
 */
private static final double MINUTES_TO_METERS = 1852d;
/**
 * the amount of minutes in one degree.
 */
private static final double DEGREE_TO_MINUTES = 60d;


/**
 * This method extrapolates the endpoint of a movement with a given length from a given starting point using a given
 * course.
 *
 * @param startPointLat the latitude of the starting point in degrees, must not be {@link Double#NaN}.
 * @param startPointLon the longitude of the starting point in degrees, must not be {@link Double#NaN}.
 * @param course        the course to be used for extrapolation in degrees, must not be {@link Double#NaN}.
 * @param distance      the distance to be extrapolated in meters, must not be {@link Double#NaN}.
 *
 * @return the extrapolated point.
 */
public static Point extrapolate(final double startPointLat, final double startPointLon, final double course,
                                final double distance) {
    //
    //lat =asin(sin(lat1)*cos(d)+cos(lat1)*sin(d)*cos(tc))
    //dlon=atan2(sin(tc)*sin(d)*cos(lat1),cos(d)-sin(lat1)*sin(lat))
    //lon=mod( lon1+dlon +pi,2*pi )-pi
    //
    // where:
    // lat1,lon1  -start pointi n radians
    // d          - distance in radians Deg2Rad(nm/60)
    // tc         - course in radians

    final double crs = Math.toRadians(course);
    final double d12 = Math.toRadians(distance / MINUTES_TO_METERS / DEGREE_TO_MINUTES);

    final double lat1 = Math.toRadians(startPointLat);
    final double lon1 = Math.toRadians(startPointLon);

    final double lat = Math.asin(Math.sin(lat1) * Math.cos(d12)
        + Math.cos(lat1) * Math.sin(d12) * Math.cos(crs));
    final double dlon = Math.atan2(Math.sin(crs) * Math.sin(d12) * Math.cos(lat1),
        Math.cos(d12) - Math.sin(lat1) * Math.sin(lat));
    final double lon = (lon1 + dlon + Math.PI) % (2 * Math.PI) - Math.PI;

    return new Point(Math.toDegrees(lat), Math.toDegrees(lon));
}

/**
 * calculates the length of one degree of longitude at the given latitude.
 *
 * @param latitude the latitude to calculate the longitude distance for, must not be {@link Double#NaN}.
 *
 * @return the length of one degree of longitude at the given latitude in meters.
 */
public static double longitudeDistanceAtLatitude(final double latitude) {

    final double longitudeDistanceScaleForCurrentLatitude = Math.cos(Math.toRadians(latitude));
    return DEGREE_DISTANCE_AT_EQUATOR * longitudeDistanceScaleForCurrentLatitude;
}

Answer 2

This isn't really an answer, but the comments box is too short for what I want to post, and this result came up quite high when I was googling around for an answer. BertNase code above is good, and I'm using it. However, there's some weirdness around the edge cases. I'm not 100% sure the code is wrong, as I'm still learning geo stuff, but I'm adding the parameters from my junit test case I wrote around it. For example longitude goes from 180 to -90 when I move south for 100m (case 10)

/*0*/   { inputOf(0.0, 0.0), NORTH, shouldGiveAnswerOf(0.0009, 0.0) },
/*1*/   { inputOf(0.0, 0.0), SOUTH, shouldGiveAnswerOf(-0.0009, 0.0) },
/*2*/   { inputOf(0.0, 0.0), WEST, shouldGiveAnswerOf(0.0, -0.0009) },
/*3*/   { inputOf(0.0, 0.0), EAST, shouldGiveAnswerOf(0.0, 0.0009) },

/*4*/   { inputOf(90.0, 180.0), NORTH, shouldGiveAnswerOf(89.9991, -180.0) },
/*5*/   { inputOf(0.0, 180.0), NORTH, shouldGiveAnswerOf(0.0009, -180.0) },
/*6*/   { inputOf(-90.0, 180.0), NORTH, shouldGiveAnswerOf(-89.9991, -180.0) },
/*7*/   { inputOf(90.0, -180.0), NORTH, shouldGiveAnswerOf(89.9991, -180.0) },
/*8*/   { inputOf(0.0, -180.0), NORTH, shouldGiveAnswerOf(0.0009, -180.0) },
/*9*/   { inputOf(-90.0, -180.0), NORTH, shouldGiveAnswerOf(-89.9991, -180) },

/*10*/  { inputOf(90.0, 180.0), SOUTH, shouldGiveAnswerOf(89.9991, -90.0) },
/*11*/  { inputOf(0.0, 180.0), SOUTH, shouldGiveAnswerOf(-0.0009, -180.0) },
/*12*/  { inputOf(-90.0, 180.0), SOUTH, shouldGiveAnswerOf(-89.9991, -90.0) },
/*13*/  { inputOf(90.0, -180.0), SOUTH, shouldGiveAnswerOf(89.9991, -90.0) },
/*14*/  { inputOf(0.0, -180.0), SOUTH, shouldGiveAnswerOf(-0.0009, -180.0) },
/*15*/  { inputOf(-90.0, -180.0), SOUTH, shouldGiveAnswerOf(-89.9991, -90) },

/*16*/  { inputOf(90.0, 180.0), EAST, shouldGiveAnswerOf(89.9991, -90.0) },
/*17*/  { inputOf(0.0, 180.0), EAST, shouldGiveAnswerOf(0.0, -179.9991) },
/*18*/  { inputOf(-90.0, 180.0), EAST, shouldGiveAnswerOf(-89.9991, -90.0) },
/*19*/  { inputOf(90.0, -180.0), EAST, shouldGiveAnswerOf(89.9991, -90.0) },
/*20*/  { inputOf(0.0, -180.0), EAST, shouldGiveAnswerOf(0.0, -179.9991) },
/*21*/  { inputOf(-90.0, -180.0), EAST, shouldGiveAnswerOf(-89.9991, -90) },

/*22*/  { inputOf(10.0, 5.0), NORTH, shouldGiveAnswerOf(10.0009, 5.0) },
/*23*/  { inputOf(10.0, 5.0), SOUTH, shouldGiveAnswerOf(9.9991, 5.0) },
/*24*/  { inputOf(10.0, 5.0), WEST, shouldGiveAnswerOf(10.0, 4.999086) },
/*25*/  { inputOf(10.0, 5.0), EAST, shouldGiveAnswerOf(10.0, 5.000914) },

/*26*/  { inputOf(10.0, 5.0), NORTH_EAST, shouldGiveAnswerOf(10.000636, 5.000646) },